Arithmetic Structure in Dense Sets

这个项目主要关注数论、组合学和遍历理论中的三个不同的问题。这包括关于Szemerédi关于算术级数(都是相等间隔的数字序列，如4、6、8和10)的推广的加法组合学方面的工作，该定理非正式地说，任何足够大的整数集合都包含一个长的算术级数。确定“足够大”有多大是加法组合学的核心问题。研究人员将研究这一问题的版本，涉及比算术级数更复杂的模式，然后使用所开发的结果和技术在遍历理论中的一个相关问题上取得进展。调查人员还将研究整数距离集的大小和结构，整数距离集是两两距离都是整数的点的集合。这一奖项将支持本科生夏季在表示理论和加性组合学方面的研究，也将支持研究生的培训。更具体地说，研究人员将在她之前工作的基础上，对缺乏不同次数多项式级数的整数子集和缺乏特定四点配置的有限域上的向量空间子集的量化界限进行研究，以解决更一般的多项式、多维和多维多项式配置。不同次数多项式构形的结果将被用于改进遍历理论中的Furstenberg-Berelson-Leibman猜想，该猜想涉及某些非常规遍历平均的逐点几乎处处收敛。她还将研究整数距离集的大小和结构，无论是在欧几里得平面上，还是在更高的维度上，通过将它们编码为某些变种上的有理点的子集，然后研究这些变种。这位研究人员将与她的本科生一起研究对称群特征标表中条目的分布以及高阶傅立叶分析中的一些算法问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project focuses primarily on three different problems in number theory, combinatorics, and ergodic theory. This includes work in additive combinatorics concerning generalizations of Szemerédi's theorem on arithmetic progressions (sequences of numbers that are all equally spaced, like 4, 6, 8, and 10), which, informally, says that any sufficiently large collection of whole numbers contains a long arithmetic progression. It is a central problem in additive combinatorics to determine how large "sufficiently large" is. The investigator will study versions of this question involving more complicated patterns than arithmetic progressions, and then use the results and techniques developed to make progress on a related problem in ergodic theory. The investigator will also study the size and structure of integer distance sets, which are sets of points whose pairwise distances are all whole numbers. This award will support undergraduate summer research on representation theory and additive combinatorics, and also support the training of graduate students.More specifically, the investigator will build on her previous work on quantitative bounds for subsets of the integers lacking polynomial progressions of distinct degrees and for subsets of vector spaces over finite fields lacking a certain four-point configuration to tackle more general polynomial, multidimensional, and multidimensional polynomial configurations. The results for multidimensional polynomial configurations of distinct degree will then be used to make progress on the Furstenberg--Bergelson--Leibman conjecture in ergodic theory, which concerns the pointwise almost everywhere convergence of certain nonconventional ergodic averages. She will also investigate the size and structure of integer distance sets, in both the Euclidean plane and in higher dimensions, by encoding them as subsets of rational points on certain families of varieties and then studying these varieties. With her undergraduate students, the investigator will study the distribution of entries in the character tables of symmetric groups and some algorithmic problems in higher-order Fourier analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.